Levy processes in inverse problems

Levy processes are a family of stationary stochastic processes that can be used to model systems close to equilibrium and are different from the Wiener process due to the introduction of jumps in the time series. Systems that exhibit jump behavior such as chaotic systems, turbulent systems, and diffusion on fractals may also be modeled using a Levy process. In measurement systems, the central limit theorem may not apply and a more general central limit theorem may describe the time series data of the noise better. For these reasons it is important to be able to classify and filter Levy processes from time series data, and in many systems an automatic classification and filtering algorithm would improve detection and classification algorithms. The classification of Levy processes will be investigated using the continuous wavelet transform. The continuous wavelet transform is effective at detecting edges or jumps in the data, and their edge detection ability will be used to classify the Levy process. The stable Levy processes are a subset of the Levy processes that are the limit process of a generalized central limit theorem, and the classification of Levy processes are limited to the stable processes in this work.; Filtering of Levy noise in ultrasound data is investigated using the discrete wavelet transform. An information theoretic technique introduced for the detection of gravitational waves is used to determine a best wavelet filter for ultrasound data. The results of this method in grain noise is discussed, and then stable Levy noise is introduced to an ultrasound signal and a best wavelet filter is found using the same information theoretic ideas. It was found that for stable Levy processes, a smooth filter will filter out the noise better.; The Feynman path integral and the Feynman-Kac formula has been used to study the Wiener process and other stochastic processes. In most measurement systems, Levy process will be confined in some manner, and therefore it is important to understand properties of the confined system. The Feynman functional is a generalization of the Feynman integral that can be used to study confined Levy processes. Semigroup theory is used to obtain perturbation series expansions of the Levy processes, and previous results on perturbations of Levy processes are obtained using the Feynman Functional method. The Ornstein-Uhlenbechk-Levy process is derived using the Feynman functional method.